Being biodegradable doesn't mean it can't cause harm before breaking down.
Once diluted, bleach starts reacting and breaking down. Throw in the dirt of the laundry, and it should all be neutralized by the time you pump it onto the lawn.

The potential risk, which I admit is tiny for your operation alone, isn't to the lawn. The grey water from most systems ends up going to watering various plants, but they do so in a controlled manner by having the water be applied by a soaker hoses or a trickle system, or directly to the soil and roots.

Applying all that water in one shot to the soil surface in a short period of time could lead to some of it running off. Even if it doesn't, the laundry chemicals and fertilizer will remain on the surface to some degree, and can get washed off with the next rain.

I don't see what you are doing causing harm, but if all your neighbors started doing it, the local creek might not like it.

You could set up a more controlled outflow with what you already have. Just move one rain barrel to outside, and pump the water into it. Open its tap just a crack so it takes hours to drain. A soaker or trickle hose can be used to direct to water to a bed or better distribute it for the lawn, but with a filtering or a settling tank, the sediment from the laundry will eventually clog it.

I found a downfall to my washing machine recycling program. Since I have to have the barrels in the house. Washer is in the basement. Don't want to depend on it's pump to pump up one floor to the outside.
Well... with the barrels in the house. The dirt accumulates in the bottom. It starts smelling bad. I had to get the barrels outside. Doh!
At least I have a very efficient washer.
Now to plan 2 with the yard. A natural yard. Wildflowers, wild bushes, Trees, grasses and all. Ones that make it well in this environment. I'll restrict it to three islands in the middle of the yard. So it will look nice. Have short well maintained grass framing all of it.